Energy-based characterisation of large-scale coherent structures in turbulent pipe flows

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Large-scale coherent structures in incompressible turbulent pipe flow are studied for a wide range of Reynolds numbers ( $Re_\tau =180, 550, 1000, 2000$ and $5200$ ). Employing the Karhunen–Loève decomposition and a novel approach based on the Voronoi diagram, we identify and classify statistically coherent structures based on their location, dimensions and $Re_{\tau }$ . With increasing $Re_{\tau }$ , two distinct classes of structures become more energetic, namely wall-attached and detached eddies. The Voronoi methodology is shown to delineate these two classes without the need for specific criteria or thresholds. At the highest $Re_{\tau }$ , the attached eddies scale linearly with the wall-normal distance with a slope of approximately $l_y\sim 1.2y/R$ , while the detached eddies remain constant at the size of $l_y \approx 0.26R$ , with a progressive shift towards the pipe centre. We extract these two classes of structures and describe their spatial characteristics, including radial size, helix angle and azimuthal self-similarity. The spatial distribution could help explain the differences in mean velocity between pipe and channel flows, as well as in modelling large and very-large-scale motions (LSM and VLSM). In addition, a comprehensive description is provided for both wall-attached and detached structures in terms of LSM and VLSM.